Notes by RBJ on

On the Concept of Logical Consequence

by Alfred Tarski

I'm just going to briefly comment on the confused nature of this paper. I think Tarski knows its a bit confused but its still worth mentioning, particularly since this is an important topic.

The paper attempts to show that logical consequence must be characterised semantically not syntactically and then to do that. However, it only partly replaces syntactic by semantic issues, it clings onto some aspects of the old syntactic accounts, and ends up admitting that the notion of consequence that comes out depends on a somewhat arbitrary division between logical and "extra-logical" expressions.

Tarski begins his paper by showing that the notion of logical consequence differs from that of derivabilty in any formal system. His does this, firstly by citing an example of logical consequence which involves an infinity of premises and which essentially involves the semantics of the notion of natural number, and then eventually by reference to Gödel's incompleteness result.

He then proceeds to a semantic account of logical consequence. However his starting point for this is that logical consequence must still be "formal" and that this requires that consequence does not depend upon the particular non-logical symbols used. Tarski appears not to notice that his first example which he used to justify the need for a semantic account of logical consequence is excluded by this criterion.

In recent times this account of logical consequence has prevailed, and the arbitrary choice of what is or is not a logical constant has settled on those of first order logic. The effect is to make logical consequence and first order derivability coincide, and to confirm the invalidation of the example which Tarski uses to motivate the paper.

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